Script to make copies of a shape key, each one with an isolated translate channel
First point would be that shape_key_add(from_mix=True) will copy the shape key as it is, you may want to be sure that the shapekey is in effect when you copy it.

obj.active_shape_key.value = obj.active_shape_key.slider_max


Rather than changing the active shapekey for each step, get the index of each as you create them and use that when adjusting the copy. The new shapekey will be added to the end of the list so it's index will be len(key_blocks)-1

xshape_idx = len(obj.data.shape_keys.key_blocks)-1


Now the key point -

obj.data.shape_keys.key_blocks[shape_name].data[vert.index].co.x 
  = obj.active_shape_key.data[vert.index].co.x


you set obj.active_shape_key_index then you copy active_shape_key.data[vert.index] into key_blocks[shape_name].data[vert.index] that is two ways to the same data. I would suggest setting the unwanted locations to the Basis shapekey locations.

obj.data.shape_keys.key_blocks[xshape_idx].data[vert.index].co.x 
  = obj.data.shape_keys.key_blocks['Basis'].data[vert.index].co.x


I finished up with -

import bpy

context = bpy.context
scene = context.scene
obj = context.object
shape_name = obj.active_shape_key.name

## enable shapekey before copying
skey_value = obj.active_shape_key.value
obj.active_shape_key.value = obj.active_shape_key.slider_max

obj.shape_key_add(name=str(shape_name) + "_X", from_mix=True)
xshape_idx = len(obj.data.shape_keys.key_blocks)-1
obj.shape_key_add(name=str(shape_name) + "_Y", from_mix=True)
yshape_idx = len(obj.data.shape_keys.key_blocks)-1
obj.shape_key_add(name=str(shape_name) + "_Z", from_mix=True)
zshape_idx = len(obj.data.shape_keys.key_blocks)-1

## reset shapekey
obj.active_shape_key.value = skey_value

for vert in obj.data.vertices: #Isolate the translation on the X axis
    obj.data.shape_keys.key_blocks[xshape_idx].data[vert.index].co.y = obj.data.shape_keys.key_blocks['Basis'].data[vert.index].co.y
    obj.data.shape_keys.key_blocks[xshape_idx].data[vert.index].co.z = obj.data.shape_keys.key_blocks['Basis'].data[vert.index].co.z

for vert in obj.data.vertices: #Isolate the translation on the Y axis
    obj.data.shape_keys.key_blocks[yshape_idx].data[vert.index].co.x = obj.data.shape_keys.key_blocks['Basis'].data[vert.index].co.x
    obj.data.shape_keys.key_blocks[yshape_idx].data[vert.index].co.z = obj.data.shape_keys.key_blocks['Basis'].data[vert.index].co.z

for vert in obj.data.vertices: #Isolate the translation on the Z axis
    obj.data.shape_keys.key_blocks[zshape_idx].data[vert.index].co.x = obj.data.shape_keys.key_blocks['Basis'].data[vert.index].co.x
    obj.data.shape_keys.key_blocks[zshape_idx].data[vert.index].co.y = obj.data.shape_keys.key_blocks['Basis'].data[vert.index].co.y